\documentclass{article}
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\usepackage{amsmath}
\begin{document}
\begin{CJK*}{UTF8}{gbsn}

\begin{equation}
{(x_{i},y_{i})}_{i=1}^{N}
\end{equation}
\begin{equation}
L=\frac{1}{N}\sum_{i}L_{i}(f(x_{i},W),y_{i})
\end{equation}

\begin{equation}
P=
   \begin{cases}
   0 &{if}\\
   1 &{if}
   \end{cases}
\end{equation}

\begin{equation}
\begin{split}
 partone &< parttwo \\
         &= parttwo 
\end{split}
\end{equation}

\begin{equation}
\begin{split}
L_{i} &= \sum_{j \neq y_{i}}
   \begin{cases}
   0 & if \quad s_{y_{i}} \geq s_{j}+1\\
   s_{j}-s_{y_{i}}+1 & otherwise
   \end{cases} \\
   &= \sum_{j \neq y_{i}} max(0,s_{j}-s_{y_{i}}+1)
\end{split}
\end{equation}

\begin{equation}
L(W)=\frac{1}{N}\sum_{i=1}^{N}L_{i}((x_{i},W),y_{i})+\lambda R(W)
\end{equation}

\begin{equation}
R(W)=\sum_{k}\sum_{l}W_{k,l}^{2}
\end{equation}

\begin{equation}
R(W)=\sum_{k}\sum_{l}\lvert{W_{k,l}\rvert}
\end{equation}

\begin{equation}
R(W)=\sum_{k}\sum_{l}\beta W_{k,l}^{2} + \lvert{W_{k,l}\rvert}
\end{equation}

\begin{equation}
L_{i}=-\log(\frac{e^{s_{y_{i}}}}{\sum_{j}e^{s_{j}}})
\end{equation}

\begin{equation}
\frac{df(x)}{dx}=\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h}
\end{equation}

\begin{equation}
\nabla_{W}L(W)=\frac{1}{N}\nabla_{W}\sum_{i=1}^{N}L_{i}((x_{i},W),y_{i})+\lambda\nabla_{W}R(W)
\end{equation}

\begin{equation}
\begin{cases}
f(x,y,z)=(x+y)z\\
x=-2,y=5,z=4\\
q=x+y,\frac{\partial q}{\partial x}=1,\frac{\partial q}{\partial y}=1\\
f=qz,\frac{\partial f}{\partial q}=1,\frac{\partial f}{\partial z}=1\\
\frac{\partial f}{\partial x}=\frac{\partial f}{\partial q}\frac{\partial q}{\partial x},\frac{\partial f}{\partial y}=\frac{\partial f}{\partial q}\frac{\partial q}{\partial y}
\end{cases}
\end{equation}

\begin{equation}
\begin{cases}
f(w,x)=\frac{1}{1+e^{-(w_{0}x_{0}+w_{1}x_{1}+w_{2})}}\\
\sigma(x)=\frac{1}{1+e^{-x}}\\
\frac{d\sigma{x}}{dx}=\frac{e^{-x}}{(1+e^{-x})^2}=(1-\sigma(x))\sigma(x)
\end{cases}
\end{equation}

\begin{equation}
f=W_{2}max(0,W_{1}x)
\end{equation}

\begin{equation}
(N-F)/stride+1
\end{equation}

\begin{equation}
f(x)=max(0,x)
\end{equation}

\begin{equation}
f(x)=max(0.01x,x)
\end{equation}

\begin{equation}
f(x)=max(\alpha x,x)
\end{equation}

\begin{equation}
f(x) = 
   \begin{cases}
   x & if \quad x>0\\
   \alpha(exp(x)-1) & x \leq 0
   \end{cases}
\end{equation}


\begin{equation}
x_{t+1}=x_{t} -\alpha\nabla f(x_{t})
\end{equation}

\begin{equation}
\begin{cases}
v_{t+1}=\rho v_{t} +\nabla f(x_{t})\\
x_{t+1}=x_{t} -\alpha v(x_{t+1})\\
\end{cases}
\end{equation}

\begin{equation}
\alpha=\alpha_{0}e^{-kt}
\end{equation}

\begin{equation}
\alpha=\alpha_{0}/(1+kt)
\end{equation}

\begin{equation}
p_{x}(i)=p(x=i)=\frac{n_{i}}{n},0\leq i<L
\end{equation}

\begin{equation}
cdf_{x}(i)=\sum_{j=0}^{i}p_{x}(j)
\end{equation}

\begin{equation}
h(v)=round(\frac{cdf(v)-cdf_{min}}{M \times N-cdf_{min}}\times(L-1))
\end{equation}

\begin{equation}
\begin{bmatrix}
    x_{11}       & x_{12} & x_{13} & \dots & x_{1n} \\
    x_{21}       & x_{22} & x_{23} & \dots & x_{2n} \\
    \hdotsfor{5} \\
    x_{d1}       & x_{d2} & x_{d3} & \dots & x_{dn}
\end{bmatrix}
=
\begin{bmatrix}
    x_{11} & x_{12} & x_{13} & \dots  & x_{1n} \\
    x_{21} & x_{22} & x_{23} & \dots  & x_{2n} \\
    \vdots & \vdots & \vdots & \ddots & \vdots \\
    x_{d1} & x_{d2} & x_{d3} & \dots  & x_{dn}
\end{bmatrix}
\end{equation}

\begin{equation}
K
= \frac{1}{9}
\begin{bmatrix}
    1 & 1& 1 \\
    1 & 1& 1 \\
    1 & 1& 1
\end{bmatrix}
\end{equation}


\begin{equation}
\begin{cases}
s=nOctaveLayers=3\\
\sigma[0]=\sigma=1.6\\
i=[1,s+2]\\
k = 2^{\frac{1}{s}}\\
\sigma_p[i] = k^{(i-1)}*\sigma\\
\sigma_t[i] = \sigma_p[i]*k\\
\sigma[i] = \sqrt{(\sigma_t[i]*\sigma_t[i]-\sigma_p[i]*\sigma_p[i])}\\
\end{cases}
\end{equation}

\begin{equation}
\begin{cases}
k = 2^{\frac{1}{s}}=2^{\frac{1}{1.6}}=1.54\\
i=0;\\
\sigma[0]=\sigma=1.6\\
i=1;\\
\sigma_p[1] = k^{(i-1)}*\sigma=1.54^0*1.6=1.6\\
\sigma_t[1] = \sigma_p[i]*k=1.6*1.54=2.46\\
\sigma[1] = \sqrt{(\sigma_t[i]*\sigma_t[i]-\sigma_p[i]*\sigma_p[i])}=\sqrt{2.46^2-1.6^2}=1.87
\end{cases}
\end{equation}

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